The dynamics of tail-like current sheets under the influence of small-scale plasma turbulence

A. V. Runov and M. I. Pudovkin

Saint-Petersburg University, Saint-Petersburg, Russia

C.-V. Meister

Astrophysical Institute, Potsdam, Germany

Contents

Abstract

Introduction

In space physics, current sheets, separating two plasma regions which have antiparallel magnetic field components, seem to play a key role. According to the terminology of Alfvén [Alfvén, 1981], current sheets are physical active regions, the (as a rule, small-scale) processes in which determine the situation in the ambient physical passive space. In this paper, the dynamics of tail-like current sheets which possess additionally a small normal component of the magnetic field [Lee et al., 1994; Lin and Lee, 1994] is considered. Tail-like configurations occur for instance in planetary magnetospheres, stellar magnetic active regions and in current sheets of stellar plasma winds. It is assumed that characteristics of stellar flares, of planetary magnetic storms and the plasmoid formation in the interplanetary medium can be explained in this way.

During the last thirty years it became clear, that the dynamical behaviour of current sheets essentially depends on the scale of the local decrease of the effective electrical conductivity in a relatively small region of the sheets, the so-called diffusion region. According to present ideas, the effective conductivity decrease in a current sheet may be caused by plasma turbulization, that is, by the excitation of plasma waves as result of some kind of plasma instability. The waves then interact with the plasma particles, and the collision frequencies of the particles increase. The corresponding additional resistivity is called "anomalous resistivity''.

The anomalous resistivity in the current sheet results in a local current destruction associated with a restructuring of the large-scale magnetic field, and thus causes a vortex electric field. During the local destruction of the current sheet energy stored in the magnetic field is transformed in a burst-like manners into kinetic energy and heat. This seems to be the cause of such events as magnetospheric substorms and stellar flares.

A schematic description of current sheet destructions in the Earth's magnetotail neglecting small normal components was already given by Dungey in 1961 [Dungey, 1961]. Dungey proposed, that a so-called x line is formed in the region of decreased conductivity. According to his model, the plasma in the destruction region can be accelerated up to the Alfvén velocity. The simple Dungey model was investigated in detail by Petschek [1964], and is now known as Petschek model of magnetic field line reconnection. There is a set of analytical (see e.g. [Pudovkin and Semenov, 1985] and the reviews by Rijnbeek and Semenov [1993] and Parker [1994]) and numerical (see [Hautz and Scholer, 1987; Scholer et al., 1990]) solutions of the Petschek reconnection problem describing the plasma motion in the vicinity of the magnetic x line.

But the process of the x line formation itself - the earliest phase of the reconnection process - is still not well understood. The investigation of the turbulization process, theoretically as well as experimentally, is very difficult. The turbulence theory is far from being completed, and it is usually very hard to say which kind of plasma instability may develop in real situations, for example, in the magnetotail or in stellar flares [Büchner et al., 1988; Coroniti, 1985; Liperovsky and Pudovkin, 1983; Parker, 1994; Schindler, 1987]. Besides, in the models the feedback of the large-scale electromagnetic and hydrodynamic processes on the small-scale plasma processes has to be taken into account.

In tail-like current sheets intensive currents and drifts as well as magnetic stresses and plasma anisotropies can cause microturbulence. It was concluded that the two most relevant instabilities for which a steady state anomalous resistivity can be achieved in almost collisionless plasmas are the ion-acoustic (IA) and the lower-hybrid-drift (LHD) instabilities [Hoshino, 1991; Huba et al., 1977]. But in the case of the IA instability, the relative electron-ion drift velocity must be much larger than v_eT_i/T_e, which is seldom the case, and dissipation may only occur in very thin diffusion sheets of a thickness of dw_pe [Coroniti, 1985] ( T_e, T_i, v_e, w_pe, c are electron and ion temperature, electron thermal velocity, electron plasma frequency and velocity of the light in vacuum). On the contrary, the LHD instability can be very effective under the condition T_i>T_e too, but it also causes a small diffusion region of the dimension of the ion Larmor radius only, 20c/w_ped (m_i/m_e)^1/4r_Li [Coroniti, 1985]. In the earth's magnetotail LHD waves were indeed observed by ISEE satellites as low-frequency part of broadband electrostatic noise.

Beginning with Ugai and Tsuda [1977], the evolution of current sheets was investigated assuming localized anomalous resistivity profiles with a peak at the neutral sheet. Hoshino [1991] was the first who considered anomalous resistivity caused by LHD turbulence out of the neutral line. But he did not take into account a small normal component of the magnetic field.

In this paper it is assumed that the current density in a local region at the symmetry plane of a tail-like current sheet exceeds the threshold of some current-driven plasma instability. It may be, for example, the electrostatic ion-cyclotron (EIC) instability in the magnetotail, or the ion-acoustic (IA) instability in solar flares (see [Galeev and Sagdeev, 1973; Kindel and Kennel, 1971; Liperovsky and Pudovkin, 1983]. The instability gives rise to a rapid decrease of the electrical conductivity of the plasma in the centre of the current sheet, which initiates the development of a magnetic reconnection pulse. This, in turn, results in the generation of relatively intensive magnetohydrodynamic waves with secondary pressure gradients at their fronts, and then it may be supposed that the LHD instability is excited by those pressure gradients.

This instability occurs if the scale of the magnetic field gradient L_B is smaller or of the order of 10 ion Larmor radii. This means, for instance, for the tail of the Earth's magnetosphere that secondary current sheets with maximum widths of approximately 2100-2800 km and diffusion regions with maximum thicknesses of about 1000 1500 km should exist. These values are in reasonable agreement with the thicknesses of current sheets in the plasma sheet during growth phases of substorms of about 3000 km given in [Sergeev et al., 1990]. In this case, the used magnetohydrodynamic approach is applicable as the condition L_B>r_Li is fulfilled, and the Debye radius is of the order of 0.3-0.4 km.

Besides, in the last years, it was also shown by Pulkkinen et al. [1992, 1994] that rather thin (500-1000 km) and relatively intense (1-20 mA m ^-1 ) current sheets develop in the near-earth magnetotail during substorm growth phases. The thin current sheets were accompanied by small normal components B_n of the magnetic field. For these current sheets, one has L_B 1-2 r_Li, and the magnetohydrodynamic approach is not valid.

Thus, we consider the development of a tail-like current sheet under the influence of both, a strong initial resistivity pulse at the neutral sheet and anomalous resistivity caused by LHD-turbulence. The main attention is paid to the behaviour of the electric field in the diffusion region and nearby, especially during the earliest phase of the reconnection process.

Model of the Current Sheet

It is assumed that the plasma of the tail-like current sheet is described by the compressible magnetohydrodynamic equations [Hoshino, 1991]

(1)

(2)

(3)

(4)

(5)

(6)

where r, p and v are the plasma density, pressure and velocity normalized by the characteristic values r₀=g/C_s²₀ p₀, p₀=B₀²/8p, v_a=B₀/4pr₀ of the undisturbed plasma. The politropic coefficient g and initial acoustic velocity C_s0 were chosen to be equal to 5/3 and 5/3 correspondingly. B designates the magnetic field normalized by the value B₀ of the current sheet environment at zd, and A is the magnetic vector-potential, normalized by A₀=B₀/L, where L is a characteristic scale. The x axis is assumed to be directed along the current sheet, so that the electric current flows in y direction. d describes the half-thickness of the sheet in z direction. Re_m=4psv_a L/c² is the magnetic Reynolds number of a plasma with characteristic length L and characteristic velocity of the order of the Alfvén velocity v_a. The temporal variable is normalized by the specific Alfvén transport time t_a=L/v_a, s designates the electrical conductivity. The hydrodynamic viscosity is neglected in the model, and the plasma pressure is assumed to be isotropic.

The initial tail-like plasma configuration is described by the solution of the Grad-Shafranov equation

(7)

(8)

(9)

(see [Hau et al., 1989; Hesse et al., 1996; Pritchett et al., 1991].)

The resistivity is supposed to have the form

(10)

[Runov et al., 1998], where h₀ is the background resistivity, which is of the order of the numerical resistivity. The first term on the right-hand side of (10)

(11)

represents the initial pulse of anomalous resistivity, introduced to start up a local current sheet destruction. t_i is the characteristic time of the initial resistivity growth, t_c gives the resistivity pulse duration, and t_r is the relaxation time of the resistivity pulse, a^-2 b^-20.1 L. The temporal variation of h_l with t_i=t_r=0.025 t_a, and t_c=0.25 t_a, used in the calculations, is shown in Figure 1. Finally,

(12)

represents the plasma resistivity within the regions of the secondary pressure gradients produced in the course of the plasma sheet evolution, and hence it describes the feedback between the resistivity and the MHD evolution of the system. h^* may be considered as the model specification.

It is assumed that h^* is caused by the LHD-instability which is driven by diamagnetic currents associated with the pressure gradient. The instability occurs if r_Li<2 L<(m_i/m_e)^1/4r_Li is valid, where L is the scale of the pressure gradient which approximately equals the thickness of the current sheet. Under the condition (m_i/m_e)^1/4r_Li<2 L the generation of ion-cyclotron drift turbulence is possible. In the high- b plasma of the magnetic neutral sheet, LHD waves are strongly damped by the interaction with the electrons. Thus, on the contrary to the initial resistivity pulse h_l, which has a maximum at the neutral line, the anomalous resistivity h^* has its maximum relatively far from the neutral line.

In the case of rather large polarization drift velocities V_i of the ions with values above 0.18 v_i(1+T_e/T_i) [Huba et al., 1978] the LHD instability may be damped by current relaxation processes and the effective plasma resistivity is proportional to V_i² ( v_i, r_Li are the thermal velocity and the Larmor radius of the ions). V_i is proportional to the gradient of the plasma density. At V_i>3v_i saturation of the wave growth by ion trapping may occur, and the effective resistivity seems to be proportional to V_i⁴. But, for instance, in the tail of the Earth's magnetosphere, at a distance of 20-30 earth radii from the Earth, seems more likely that LHD instabilities are saturated by current relaxation. Thus, one can assume, that the electrical plasma resistivity in the turbulent region at maximum wave growth is about 10⁸ W m, that means the resistivity may increase up to four-five orders in comparison with the plasma without turbulence [Meister, 1997].

The values of h^* may be different in different models. So, Schumacher and Kliem [1997] assumed it to be proportional to the excessive current density j-j_cr in linear or quadratic dependence.

In this paper, following [Hoshino, 1991], the effective resistivity in the turbulent region is supposed to be proportional to the square of the mean magnetic field gradient (13). This magnetic-field-gradient dependence was already published in [Huba et al., 1977].

Simulation Results

The system of (1)-(13) is solved numerically using the two-step predictor-corrector Lax-Wendroff method [Scholer and Roth, 1987; Scholer et al., 1990] and the absolutely stable Dufort-Frankel scheme [Dautray and Lions, 1993]. The both methodes are of second order of precision with respect to space and time grid steps. The computation was made in the range 1 x/L10, 0 z/L1 in uniform numerical box 100 100 meshes, with cell sizes Dx/L=0.05, Dz/L=0.01. The time step was calculated from the Courant-Friedrichs-Levy stability condition [Dautray and Lions, 1993], Dt/t10^-5.

The free boundary conditions are specified at the boundaries x=0, x=10, and z=1. It means that mass flux, energy flux and magnetic field can freely exit the numerical box [Hoshino, 1991; Scholer and Roth, 1987]. The symmetry/antysymmetry conditions are specified at the z=0 boundary.

The following model parameters of the initial configuration were chosen: d=0.1 L, k=5.0, a=-3.5.

The inverse initial magnetic Reynolds number pulse (see (11) and (12)) used for the calculations is represented in the upper panel of Figure 1 as function of time. The maximum value of h_l/h₀ equals 50 in the center of the diffusion region.

In accordance with the ideas proposed in [Pudovkin et al., 1997], we consider the inductive electric field E=(1/c) ( A/ t) as the main characteristic of the current sheet evolution. In Figure 1 the temporal variation of the inductive electric field at the center of the initial diffusion region (curve 1) and on the sheet periphery (curve 2) are shown. The field E is normalized by the Alfvén electric field E_a=B₀ v_a/c. As it follows from the calculations, the inductive electric field at first rapidly increases when the local resistivity increases, and the steepeness of the E(t) curve is determined by the rate of the h_l -increase. Then, in spite of the fact that the h_l -value remains constant, the electric field intensity gradually decreases with a characteristic time scale equal to the diffusion time t_d=4ps_s l_d²/c², where s_s is the average of the electrical conductivity in the diffusion region, and l_d is the half-width of the current sheet. It should be emphasized that if the anomalous resistivity is varying slowly (w1/t_d), then the temporal variation of the inductive electric field is independent of the temporal behaviour of the resistivity and may be used for the estimate of the value of h_l.

In Figures 2, 3, and 4, the spatial evolution of the inductive electric field is presented. The calculations show that at the initial stage of the process, the inductive electric field is localized in the vicinity of the given anomalous resistivity region, and with the time the electric field, which gives the information on the disturbances in the plasma-magnetic field system, propagates as in a wave-like manner along and across the sheet. An estimations of the velocity of the electric field propagation shows that the electric field wave moves across the sheet as the fast magneto-acoustic wave with the velocity v v_f=v_a²+C_s², and as the acoustic wave with the velocity v C_s=gp/r along the sheet. It should be noted that the electric field maxima are moving from the sheet axis to the sheet periphery (see Figure 3).

At t=1.00t the electric field intensity in the vicinity of the x line is very small (Figure 4). At the same time, a rather intensive electric field goes to existing at the sheet periphery. In the case of the Earth's magnetotail, that means, that if a second resistivity pulse would occur within a relatively short time interval, it may appear together with a background electric field in the tail lobes. This seems to be an important conclusion suggesting that during substorms in the tail, there should exist electric field pulses with characteristic time scale of ~10 s, and a background electric field with smooth temporal variation forming an enveloping curve of the pulses.

In Figure 5 the contours of anomalous resistivity h^*(x,z) at t=1.00t are shown. The maxima of the anomalous resistivity at t=1.00t correspond to the maxima of the inductive electric field. It should be noted that h^* develops at the periphery of the sheet, where the plasma- b equals approximately unity. No anomalous resistivity occurs near the symmetry axis of the current sheet, and only secondary diffusion regions connected with the anomalous resistivity caused by LHD-type plasma turbulence exist.

The plasma convection in the vicinity of the magnetic field x line is demonstrated in Figure 6. The convection is quasi-hyperbolic, which is in agreement with the Petschek-type reconnection model (see [Rijnbeek and Semenov, 1993]). The two regions containing the accelerated plasma and the anomalous B_z component correspond to the so-called field reversal (FR) regions, postulated in Petschek-type models.

Figure 7 displays the plasma density variation dr=r(t+Dt)-r(t) at t=1.00 t, where dt is the time step. The figure shows, that the perturbation of the initial steady-state plasma-magnetic field configuration by the resistivity pulse in a local but finite region is followed by the generation of three types of magnetohydrodynamic wave-like disturbances. The existence of such waves was first shown numerically in [Runov and Pudovkin, 1996]. Then Semenov et al. [1997] analytically found that these waves may appear. The first type is the magnetoacoustic wave of rarefaction propagating across the magnetic field. This disturbance is characterized by small amplitude negative variations of the plasma density (dr<0) and the absolute value of the magnetic field (dB<0). The second is the fast magnetoacoustic wave of compression (dr>0, dB>0), associated with the FR region. And the third is the slow magnetoacoustic wave (dr>0, dB<0), corresponding to the rotation of the plasma convection vector (wave theory see [Lin and Lee, 1994; Polovin and Demutzkiy, 1987; Sturrock, 1994]).

As it follows from the calculations, the inductive electric field and therewith the information on the disturbances is carryed by the fast magnetoacoustic wave. This seems to be an important result, because the effects of the fast wave are often neglected in Petschek-type reconnections models. The maximum of anomalous resistivity is associated with the high magnetic pressure gradient in the transition region from the fast wave of compression, where the kinetic energy of plasma has the maximum, to the slow magnetoacoustic wave, where the magnetic energy has a minimum.

Conclusions

A very simple physical model of a tail-like current sheet with a normal component of the magnetic field was considered. Of course, real magnetospheric substorms or solar flares occur under much more complex conditions. But, nevertheless, the model allows one to formulate general conclusions, which may be useful for the analysis of spacecraft and ground based data of substorms and flares.

1. In the vicinity of the magnetic x line, after the initial resistivity pulse onset, the electric field at first rapidly increases up to a value of about E_max 0.1 E_a, and then decreases exponentially with a characteristic time scale equal to the diffusion time. At the late phase of the reconnection pulse, the resistivity is determined only by the anomalous LHD-resistivity, the feedback on which by the macroscopic magnetic field changes was taken into account.

2. Further, it was shown that the pulse of anomalous resistivity generates a series of magnetohydrodynamic waves propagating from the x line. The waves are: first, fast magnetoacoustic waves with synphase variations of plasma density and absolute value of the magnetic induction. Such waves propogate along the current sheet as a compression wave and across the sheet as a rarefaction wave. Second, slow magnetoacoustic waves with anticorrelation between the variations of plasma density and absolute value of the magnetic induction were found.

3. The numerical calculation showed that during reconnection processes the inductive electric field, which contains information on the disturbances in a plasma-magnetic field system, is carryed by the fast magnetoacoustic waves along and across the current sheet.

Acknowledgments

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